Mathematically strong subsystems of analysis with low rate of growth of provably recursive functionals
Ulrich Kohlenbach Fachbereich Mathematik J.W. Goethe–Universit¨at D–60054 Frankfurt, Germany
September 1995
Abstract This paper is the first one in a sequel of papers resulting from the authors Habilitationsschrift [22] which are devoted to determine the growth in proofs of standard parts of analysis. A ω hierarchy (GnA )n∈IN of systems of arithmetic in all finite types is introduced whose definable objects of type 1 = 0(0) correspond to the Grzegorczyk hierarchy of primitive recursive functions. ω We establish the following extraction rule for an extension of GnA by quantifier–free choice δ η AC–qf and analytical axioms Γ having the form ∀x ∃y ≤ρ sx∀z F0 (including also a ‘non– standard’ axiom F − which does not hold in the full set–theoretic model but in the strongly majorizable functionals):
ω 1 0 0 From a proof GnA +AC–qf + Γ ⊢∀u ,k ∀v ≤τ tuk∃w A0(u,k,v,w) one can extract a uniform bound Φ such that 1 0 ∀u ,k ∀v ≤τ tuk∃w ≤ ΦukA0(u,k,v,w) holds in the full set–theoretic type structure.
In case n = 2 (resp. n = 3) Φuk is a polynomial (resp. an elementary recursive function) in M ω − k, u := λx. max(u0,...,ux). In the present paper we show that for n ≥ 2, GnA +AC–qf+F proves a generalization of the binary K¨onig’s lemma yielding new conservation results since the ω conclusion of the above rule can be verified in Gmax(3,n)A in this case. In a subsequent paper we will show that many important ineffective analytical principles ω and theorems can be proved already in G2A +AC–qf+Γ for suitable Γ.
1 Introduction
This paper is the first one in a sequel of papers resulting from the authors Habilitationsschrift [22] which are devoted to determine the growth in proofs of standard parts of analysis. 0 Let U be a complete separable metric space, K a compact metric space and A ∈ Σ1. As we have elaborated in [21] many numerically interesting theorems in analysis can be transformed into sentences having the form
(1) ∀u ∈ U, k ∈ IN∀v ∈ K∃w ∈ IN A(u,k,v,w)
1 and one is interested in a uniform bound Φuk on w which does not depend on v ∈ K, i.e.
∀u ∈ U, k ∈ IN∀v ∈ K∃w ≤ Φuk A(u,k,v,w).
Quite often A is monotone with respect to w, i.e.
A(u,k,v,w1) ∧ w2 ≥ w1 → A(u,k,v,w2) and hence the bound Φuk in fact realizes ‘∃w’ (see [21] for a discussion of this phenomenon). What do we know about the rate of growth of Φ if we know that (1) is proved using certain parts of analysis? In [14],[15], [19],[20] we have developed a proof–theoretic method suited for the extraction of such bounds from proofs in analysis which guarantees the extractability of primitive recursive bounds for large parts of analysis. Moreover this method has been applied to concrete (ineffective) proofs in approximation theory yielding new a–priori estimates for numerically relevant data as constants of strong unicity and others which improve known estimates significantly (see [19],[20],[21]).
In analyzing these applications we developed in [21] a new monotone functional interpretation which has important advantages over the method from [15] and provides a particular perspicuous procedure of analyzing ineffective proofs in analysis.
The starting point for the investigation carried out in the present paper is the following problem: Whereas the general meta–theorems in [15], [19] and [21] only guarantee the existence of a primitive recursive bound Φ, the bounds which are actually obtained in our applications to approximation theory have a very low rate of growth which is polynomial (of degree ≤ 2) relatively to the growth of the data of the problem. Thus the problem arises to close the still large gap between polynomial and primitive recursive growth. Before we start to discuss this question let us note that using a suitable representation of spaces like U,X and the basic notions of real analysis, sentences (1) can be formalized in the language of arithmetic in all finite types such that (1) gets (a special case of) the following logical form1
1 0 0 (2) ∀u , k ∀v ≤τ tu k∃w A0(u, k,v,w).
1 1 1 0 0 0 Here u := u1,...,un, k := k1,...,km, t is a closed term, τ an arbitrary finite type, 1 = 0(0) and A0(u, k,v,w) a quantifier–free formula containing only the free variables u, k,v,w. ≤τ is defined pointwise. By a uniform bound we now mean a functional Φ such that
1 0 ∀u , k ∀v ≤τ tu k∃w ≤0 Φu kA0(u, k,v,w).
Again the predicate ‘uniform’ for the bound Φ refers to the fact that Φ does not depend on v. Coming back to our question from above we are interested in the determination of those parts of classical analysis, where the extractability of bounds Φ having only polynomial growth (resp. elementary recursive growth) relatively to the data is guaranteed. ω In order to address this question we introduce a hierarchy GnA of subsystems of classical arith- metic in all finite types and investigate the rate of growth caused by various analytical principles ω 1(1) ω relatively to GnA +AC–qf. The definable functionals t in GnA are of increasing order of growth:
1 ω For the weak system G2A discussed below more subtle representations than those which are used in [19] are necessary. Such representations are developed in 3 of [22] and will be published in a paper under preparation.
2 If n = 1, then tf 1x0 is bounded by a linear function in f M , x, if n = 2, then tf 1x0 is bounded by a polynomial in f M , x, if n = 3, then tf 1x0 is bounded by an elementary recursive (i.e. a (fixed) finitely iterated exponential) function in f M , x, where f M := λx0. max(f0,...,fx) and Φfx is called linear (polynomial, elementary recursive) in 1 0 0 0 1 1 f, x if ∀f , x (Φfx =0 Φ[f, x]) for a term Φ[f, x] which is built up from 0 , x ,f ,S , + (respectively 00, x0,f 1,S 1, +, and 00, x0,f 1,S1, +, , λx0,y0.xy) only. In our results the term Φ[f, x] can always be constructed.
Let us motivate this notion for the polynomial case: If Φfx is a polynomial in f 1, x0, then in particular for every polynomial p ∈ IN[x] the function λx0.Φpx can be written as a polynomial in IN[x]. Moreover there exists a polynomial q ∈ IN[x] (depending only on the term structure of Φ) such that For every polynomial p ∈ IN[x] one can construct a polynomial r ∈ IN[x] such that 1 0 ∀f f ≤1 p → ∀x (Φfx ≤0 r(x)) and deg(r) ≤ q(deg(p)). 1(1) ω M M Since every closed term t in G2A is bounded by a polynomial Φf x in f , x and f ≤1 p → M f ≤1 p (since p is monotone) this also holds for tfx instead of Φfx.
k
1 0 (0) . . . (0) ω In particular every closed term t (t ) of G2A is bounded by a polynomial pt ∈ IN[x] (resp. a polynomial pt ∈ IN[x1,...,xk]). 1 ω n For general n ∈ IN, n ≥ 1, every closed term t of GnA is bounded by some function ft ∈E where En denotes the n–th level of the Grzegorczyk hierarchy.
ω It turns out that many basic concepts of real analysis can be defined already in G2A : e.g. rational numbers, real numbers (with their usual arithmetical operations and inequality rela- tions), d–tuples of real numbers (for every fixed d), sequences and series of reals, continuous functions F : IRd → IR and uniformly continuous functions F : [a,b]d → IR, the supremum of F ∈ C([a,b]d, IR) on [a,b]d, the Riemann integral of F ∈ C[a,b]. Furthermore the trigonometric functions sin, cos, tan, arcsin, arccos, arctan and π as well as the restriction expk (lnk) of the expo- ω nential function (logarithm) to [−k, k] for every fixed number k can be introduced in G2A (The ω unrestricted functions exp and ln can be defined in G3A ). ω G2A +AC–qf proves many of the basic properties of these objects. − − In this paper we determine the growth of extactable bounds Φ for GnA+AC–qf+F , where F ω is a certain analytical axiom which allows (relatively to G2A +AC–qf) very short and perspicuous proofs of fundamental theorems of analysis as e.g.
• every pointwise continuous function f : [0, 1]d → IR is uniformly continuous and possesses a modulus of uniform continuity
• the attainment of the maximum value of f ∈ C([0, 1]d, IR) on [0, 1]d
• the sequential form of the Heine–Borel covering property for [0, 1]d
3 • Dini’s theorem together with a modulus of uniform convergence • the existence of a uniformly continuous inverse function for every strictly increasing continuous function f : [0, 1] → IR.
In particular we show the following: δ τ Let ∆ be a set of sentences having the form ∀x ∃y ≤ρ sx∀z B0 (B0 quantifier–free). Then the following rule holds:
ω − 1 0 0 From a given proof GnA +AC–qf+∆ + F ⊢ ∀u , k ∀v ≤τ tu k∃w A0(u, k,v,w) (∗) one can extract a uniform bound Φ such that G Aω + ∆+ b-AC ⊢ ∀u1, k0∀v ≤ tu k∃w ≤ Φu k A (u, k,v,w), max(n,3) i τ 0 0 where
Φu k is a polynomial in uM , k if n =2 Φu k is an elementary recursive function in uM , k if n =3.
Here b–AC denotes the schema
δ,ρ ρδ δ δ,ρ (b–AC ) : ∀Z ∀x ∃y ≤ρ Zx A(x,y,Z) → ∃Y ≤ρδ Z∀xA(x, Y x, Z) , b–AC := (b–AC ) . δ,ρ∈T
If ∆ consists of sentences B which hold in the full set–theoretic type Sω (where set–theoretic refers to say ZFC) then one can conclude that
ω 1 0 S |= ∀u , k ∀v ≤τ tu k∃w ≤0 Φu k A0(u, k,v,w), i.e. the bound Φ is verified in the full set–theoretic model although F − is not valid in Sω but only in the model Mω of so–called strongly majorizable functionals (see 4). ω (If ∆ = ∅ then we have a verification already in Gmax(n,3)Ai , i.e. without b–AC). In a subsequent paper we will show that substantial parts of classical analysis can be developed in ω − G2A +AC–qf+∆ + F for suitable ∆ or if the proof uses functions having e.g. exponential growth ω − in G3A +AC–qf+∆ + F (In the later case one obtains bounds which are polynomial relatively to these exponential functions. If these functions are not used iterated in the given proof one gets bounds having essentially simple exponential growth instead of being merely elementary recursive; see remark 3.2.6 for a discussion of this point), e.g. in addition to the theorems mentioned above we have
• the fundamental theorem of calculus • Fej´er’s theorem on the uniform approximation of 2π–periodic continuous functions by trigono- metric polynomials
• the equivalence (local and global) of ε-δ–continuity and sequential continuity of F : IR → IR
• Mean value theorems for differentiation and integrals • Cauchy–Peano existence theorem for ordinary differential equations
• Brouwer’s fixed point theorem for continuous functions F : [0, 1]d → [0, 1]d.
4 In a further paper we will consider the growth caused by single sequences of instances of principles like • the convergence of bounded monotone sequences of real numbers • the existence of a greatest lower bound for sequences of reals which are bounded from below
• the Bolzano–Weierstra property for bounded sequences in IRd • the Arzel`a–Ascoli lemma.
ω − relatively to G2/3A +AC–qf+∆ + F . Whereas the full versions of these principles are equivalent ω to the schema of arithmetical comprehension (provably in G2A ) and thus prove the totality of every α(< ε0)–recursive function, it turns out that single sequences of instances (which however may depend on the parameters of the conclusion) of these principles contribute to the growth of bounds at most by certain primitive recursive functionals (in the sense of [11],[12]). There are even important special cases where their contribution is only polynomial. In contrast to this, single instances of the principle of • the existence of the limit superior of bounded sequences in IR may contribute a growth of the Ackermann type. For these results a combination of the techniques developed in this paper with a new method of eliminating Skolem functions for monotone formulas will be used. The present paper is devoted mainly to establish (∗). Furthermore as a proof–theoretic application of (∗) we obtain (see 4 below) conservation results for a generalization WKLseq of the binary K¨onig’s 2 lemma WKL to sequences of trees: We give a new formulation WKL(seq) of WKL(seq) which avoids ω ω the need of a coding functional Φ fx = fx (which is not available in G2A but only in GnA for ω n ≥ 3) by the use of functionals of higher type (relatively to G3A both formulations turn out to 2 ω − 1,0 0,1 be equivalent). WKLseq is provable in G2A + F +AC –qf+AC –qf. Thus (∗) also applies to 2 proofs using WKLseq and in particular we obtain the following rule
ω 2 0 0 From a proof G2A +AC–qf+WKLseq ⊢ ∀u ∀v ≤τ tu∃w A0(u,v,w) one can extract constants k,c1,c2 ∈ IN such that G Aω ⊢ ∀u0∀v ≤ tu∃w ≤ c uk + c A (u,v,w). 3 i τ 0 1 2 0 ω Finally let us emphasize that our systems based on G2A +AC–qf must not be confused with systems ω of ‘feasible analysis’ as defined e.g. (in a second–order setting) in [6]. In G2A one can define for instance functionals which compute 1 f(x)dx or sup f(x) for uniformly continuous functions 0 x∈[0,1] f ∈ C[0, 1] (endowed with a modulus of uniform continuity) although these notions are not (known to be) feasible (see [13]). Thus the formula A in (1) above may involve terms like 1 f(x)dx or 0 supx∈[0,1] f(x) and it is only by this fact that (1) covers many theorems in analysis. Nevertheless we obtain polynomial bounds p ∈ IN[k] such that ∀k ∈ IN∀v ∈ K∃w ≤ p(k) A(k,v,w) from proofs of ω − ∀k ∈ IN∀v ∈ K∃w A(k,v,w) in G2A +AC–qf+∆+F (and in the presence of u ∈ U polynomials in uM ). By monotonicity of A in w these bounds usually yield realizations for ∃w (which in particular are computable in polynomial time and therefore ‘feasible’ since p is a polynomial!). Acknowledgment: I am grateful to Prof. H. Luckhardt who encouraged me to investigate proof– theoretically substantial subsystems of analysis producing mathematical bounds of low – in particular polynomial – growth.
5 2 Subsystems of primitive recursive arithmetic in all finite types
2.1 Classical and intuitionistic predicate logic PLω and HLω in the lan- guage of all finite types The set T of all finite types is defined inductively by
(i) 0 ∈ T and (ii) ρ, τ ∈ T ⇒ τ(ρ) ∈ T.
Terms which denote a natural number have type 0. Elements of type τ(ρ) are functions which map objects of type ρ to objects of type τ. The set P ⊂ T of pure types is defined by (i) 0 ∈ P and (ii) ρ ∈ P ⇒ 0(ρ) ∈ P. Brackets whose occurrences are uniquely determined are often omitted, e.g. we write 0(00) instead of 0(0(0)). Furthermore we write for short τρk ...ρ1 instead of τ(ρk) . . . (ρ1). Pure types can be represented by natural numbers: 0(n) := n+1. The types 0, 00, 0(00), 0(0(00)) . . . are so represented by 0, 1, 2, 3 . . .. For arbitrary types ρ ∈ T the degree of ρ (for short deg(ρ) ) is defined by deg(0) := 0 and deg(τ(ρ)) := max(deg(τ),deg(ρ) + 1). For pure types the degree is just the number which represents this type. Functions having a type whose degree is > 1 are usually called functionals. The language L(HLω) of HLω contains variables xρ,yρ,zρ,... for each type ρ ∈ T together with corresponding quantifiers ∀xρ, ∃yρ as well as the logical constants ∧, ∨, → and an equality relation
=0 between objects of type 0. Furthermore we have a propositional constant ⊥ (‘falsum’). Negation ω as a defined notion: ¬A :≡ A → ⊥. Finally L(HL ) contains ‘logical’ combinators Πρ,τ and Σδ,ρ,τ of type ρτρ and τδ(ρδ)(τρδ) for all ρ,τ,δ ∈ T. HLω has the usual axioms and rules of intuitionistic predicate logic (for all sorts of variables) plus the equality axioms for =0 (e.g. see [34] ). Equations s =ρ t between terms of higher type ρ =0ρk ...ρ1 ρ1 ρk are abbreviations for the formulas ∀x1 ,...,xk (sx1 ...xk =0 tx1 ...xk). Πρ,τ , Σδ,ρ,τ are characterized by the corresponding axioms of typed combinatory logic:
ρ τ Πρ,τ x y =ρ x and Σδ,ρ,τ xyz =τ xz(yz) where x ∈ τρδ,y ∈ ρδ,z ∈ δ. Furthermore we have the following quantifier–free rule of extensionality
A0 → s =ρ t QF–ER : , where A0 is quantifier–free. A0 → r[s]=τ r[t] Classical predicate logic in all finite types PLω results if the tertium–non–datur schema A ∨ ¬A is added to HLω. The enrichment of HLω (resp. PLω ) obtained by adding the extensionality axiom
ρ ρ τρ (Eρ) : ∀x ,y ,z (x =ρ y → zx =τ zy) for every type ρ is denoted by E–HLω (resp. E–PLω).
ρ τ Remark 2.1.1 Using Πρ,τ and Σδ,ρ,τ one defines (e.g. as in [34] ) λ–terms λx .t [x] for each term tτ [xρ] such that ω ρ τ ρ ′ ρ τ HL ⊢ λx .t [x] s =τ t[s]. In particular one can define a combinator Π = λx ,y .y such that ρ,τ ′ ρ τ ′ Πρ,τ x y =τ y (E.g. take Π := Π(ΣΠΠ) for Σ, Π of suitable types).
Notational convention: Throughout this paper A0,B0, C0,... always denote quantifier–free for- mulas.
6 2.2 Subsystems of arithmetic in all finite types corresponding to the Grzegorczyk hierarchy In the following we extend PLω and HLω by adding certain computable functionals and universal axioms including the schema of quantifier–free induction. The following definition from [28] is a variant of a definition due to [1] and can be used for a perspicuous definition of the well–known Grzegorczyk hierarchy from [9] (see def.2.2.27 ).
Definition 2.2.1 For each n ∈ IN we define (by recursion on n from the outside) the n-th branch of the Ackermann function An : IN × IN → IN by
′ ′ A0(x, y) := y (Here and in the following x stands for the successor Sx of x),
x, if n =0 An+1(x, 0) := 0, if n =1 . 1, if n ≥ 2, ′ An+1(x, y ) := An(x, An+1(x, y))
..x y x. Remark 2.2.2 1) A1(x, y)= x + y, A2(x, y)= x y, A3(x, y)= x , A4(x, y)= x (y times).
2) For each fixed n ∈ IN the function An is primitive recursive. But: A(x) := Ax(x, x) is not primitive recursive.
ω We now define the Grzegorczyk arithmetic GnA of level n ≥ 1 in all finite types and their ω intuitionistic variant GnAi :
ω ω 00 L(GnA ) is defined as the extension of L(PL) ) by the addition of function constants S (successor), 000 000 000 000 001 001 001 max0 , min0 , A0 ,...,An and functional constants Φ1 ,..., Φn , µb (bounded µ–operator),
R˜ρ ∈ ρ(ρ0)(ρ00)ρ0 (for each ρ ∈ T). Furthermore we have a predicate symbol ≤0. ω ω In addition to the axioms and rules of PL the theory GnA contains the following:
1) ≤0–axioms: x ≤0 x, x ≤0 y ∨ y ≤0 x, x ≤0 y ∧ y ≤0 z → x ≤0 z, x ≤0 y ∧ y ≤0 x ↔ x =0 y.
2) S–axioms: Sx =0 Sy → x =0 y, ¬0=0 Sx, x ≤0 Sx.
3) (max) : max0(x, y) ≥0 x, max0(x, y) ≥0 y, max0(x, y)=0 x ∨ max0(x, y)=0 y.
4) (min) : min0(x, y) ≤0 x, min0(x, y) ≤0 y, min0(x, y)=0 x ∨ min0(x, y)=0 y.
5) The defining recursion equations for A0,...,An from the definition 2.2.1 above.
6) Defining recursion equations for Φ1,..., Φn:
Φif0=0 f0 ′ ′ Φifx =0 Ai−1(fx , Φifx) for i ≥ 2 and
Φ1f0=0 f0 ′ ′ Φ1fx =0 max0(fx , Φ1fx). 7 (For i ≥ 2, Φi is the iteration of the (i − 1)-th branch Ai−1 of the Ackermann function on the f–values f0,...,fx for variable x).
000 y ≤0 x ∧ f xy =0 0 → fx(µbfx)=0 0, 7) (µb) : y <0 µbfx → fxy =0, µ fx = 0 ∨ (fx(µ fx)= 0 ∧ µ fx ≤ x) b 0 b 0 b 0 (These axioms express that µbfx = min y ≤0 x(fxy =0 0) if such an y ≤ x exists and = 0 otherwise).
8) Defining recursion equations for R˜ρ (bounded and predicative recursion, since only type–0– values are used in the recursion):
˜ Rρ0yzvw =0 yw ′ R˜ρx yzvw =0 min0(z(R˜ρxyzvw)xw, vxw), ρ1 ρk where y ∈ ρ =0ρk ...ρ1, w = w1 ...wk , z ∈ ρ00, v ∈ ρ0.
IN (ININ) 9) All IN, IN , IN –true purely universal sentences ∀xA0(x), where x is a tuple of variables whose types have a degree ≤ 2, i.e. all such sentences which are true in the full type–structure Sω of all set–theoretic functionals, where ‘set–theoretic’ refers to say ZFC (The constants introduced so far have an interpretation in Sω which is uniquely determined by the axioms 1)–8). By this interpretation Sω becomes a model of the theory axiomatized by 1)–8). It is this model we refer to if we speak of ‘truth’ in Sω).
ω ω GnAi is the variant of GnA with intuitionistic logic only. ω ω ω If we add (E) = ρ {(Eρ)} to GnA ,GnAi we obtain theories which are denoted by E–GnA , ω ω ω E–GnAi . GnR denotes the set of all closed terms of GnA .
Remark 2.2.3 1) The functionals Φ1, Φ2 and Φ3 have the following meaning: x x Φ1fx = max(f0,f1,...,fx), Φ2fx = fy, Φ3fx = fy. y=0 y=0 ω 2) Our definition of GnA contains some redundances (which however we want to remain for
greater flexibility of our language): E.g. Φi (i > 1) can be defined from Ai, R,˜ min0 and Φ1: M M With f := λx.Φ1fx, 2.2.18 below implies Φifx ≤ Ai(f x +1, x + 1). Hence Φi can be M defined by R˜ using Ai(f x +1, x + 1) as boundary function v. 3) The axiom of quantifier–free induction
1 0 ′ (1) ∀f , x f0=0 0 ∧ ∀y < x(fy =0 0 → fy =0 0) → fx =0 0 1 0 can be expressed as an universal sentence ∀f , x A0 by prop.2.2.6 below and thus is an axiom ω of GnAi . (1) implies every instance (with parameters of arbitrary type) of the schema of quantifier–free induction
0 ′ QF–IA : ∀x A0(0) ∧ ∀y < x(A0(y) → A0(y )) → A0(x)
since again by prop.2.2.6 there exists a term t such that tx =0 0 ↔ A0(x): QF–IA now follows from (1) applied to f := t.
8 4) Because of the axioms in 9), our theories are not recursively enumerable. The motivation for the addition of these sentences as axioms is two–fold: (i) As G. Kreisel has pointed out in various papers, proofs of IN–true universal lemmas have no impact on bounds extracted from proofs using such lemmas. For the methods we use for the extraction of bounds (e.g. our monotone functional interpretation) this applies even for ρ arbitrary universal sentences ∀x A0 where ρ may be an arbitrary type. Taking such sentences as axioms usually simplifies the process of the extraction of bounds enormously. The reason for our restriction to those sentences for which ρ ≤ 2 is that on some places in this paper we deal with principles which are valid only in the type structure Mω of the so–called strongly majorizable functionals (see 4 below) but not in the full type structure Sω of all set–theoretic functionals. Since both type structures coincide up to type 1 and for the type 2 the inclusion ω ω ω ρ ω ρ M2 ⊂ S2 holds, the implication S |= ∀x A0 ⇒ M |= ∀x A0 holds if ρ ≤ 2. The same is true if we replace Mω by the type structure ECF of all extensional continuous functionals over ININ (see [34] for details on ECF). (ii) Many important primitive recursive functions such as sg, sg, |x − y| and so on are already ω definable in G1A . However the usual proofs for their characteristic properties (which can be ω expressed as universal sentences) often make use of functions which are not definable in G1A (as e.g. x y). Thus we would have to carry out the boring details of a proof for these properties ω in G1A .
ω Using R˜0 the following primitive recursive functions can be defined easily in G1A :
Definition 2.2.4
prd(0) =0 0 1) ′ prd(x )=0 x (predecessor),
sg(0) =0 0 sg(0) =0 1 (1 := S0) 2) ′ ′ sg(x )=0 1, sg(x )=0 0,
x − 0=0 x 3) ′ x − y =0 prd(x − y), 4) |x − y| =0 max(x − y,y − x) (symmetrical difference),
5) ε(x, y)=0 sg(|x − y|) (characteristic function for =0),
6) δ(x, y)=0 sg(|x − y|) (characteristic function for =).
ω Remark 2.2.5 Because of the universal axioms in 9), the theory G1Ai proves the usual properties of the functions max, min,prd,sg, sg, − , |x − y|,ε and δ, e.g. sg(x)=0 ↔ x =0, sg(x)=0 ↔ x =0, sg(x) ≤ 1, sg(x) ≤ 1, prd(x) ≤ x, |x − y| =0 ↔ x = y, x =0 ∨ x = S(prd(x)), max(x, y)=0 ↔ x =0 ∧ y =0, min(x, y)=0 ↔ x =0 ∨ y =0, max(x, y)=0 y ↔ x ≤0 y.
9 ω Proposition: 2.2.6 Let n be ≥ 1. For each formula A ∈ L(GnA ) which contains no quantifiers ω except for bounded quantifiers of type 0 one can construct a closed term tA in GnA such that
ω ρ1 ρk GnA ⊢ ∀x ,...,x tAx1 ...xk =0 0 ↔ A(x1,...,xk) , i 1 k where x1,...,xk are all the free variables of A.
Proof: Induction on the logical structure of A using the remark above. Bounded quantifiers are captured by µb:
ω (µb) GnAi ⊢ ∃y ≤0 xA(x,y,a) ↔ A(x, µb(λx, y.tAxya, x),a).
ω ρ1 ρk Proposition: 2.2.7 Let n ≥ 1, A0(x) ∈ L(GnA ), where x = x1 ...xk are all free variables of
0ρk ...ρ1 0ρk ...ρ1 ω 0ρk...ρ1 A0, and t1 ,t2 are closed terms of GnA . Then there exists a closed term Φ in ω GnA such that
ω t1x, if A0(x) GnAi ⊢ ∀x Φx =0 t2x, if ¬A0(x). Proof: ′ 0 0 ′′ 0 ˜ ′ ′′ Define t2 := λy ,u .t2, t2 := λu .t2. One easily verifies that Φ := λx.Rρ(tA0 x)t1t2t2 x with tA0 as in the previous proposition and ρ =0ρk ...ρ1 fulfils our claim. Definition 2.2.8 (and lemma) For n ≥ 2 we can define the surjective Cantor pairing function j 2 ω (’diagonal counting from below’) with its projections in GnR :
2 2 0 0 min u ≤0 (x + y) +3x + y[2u =0 (x + y) +3x + y] if existent j(x ,y ) := 00, otherwise, 3 j1z := min x ≤0 z[∃y ≤ z(j(x, y)= z)],
j2z := min y ≤0 z[∃x ≤ z(j(x, y)= z)].
Using j, j1, j2 we can define a coding of k–tuples for every fixed number k by
1 2 k+1 k ν (x0) := x0, ν (x0, x1) := j(x0, x1), ν (x0,...,xk) := j(x0, ν (x1,...,xk)), j ◦ (j )i−1(x), if 1 ≤ i < k k 1 2 νi (x1,...,xk) := (if k> 1) k−1 (j2) (x), if 1
2 k+1 := 0, x0,...,xk := S(ν (k, ν (x0,...,xk))).
ω Using R˜ one can define functions lth, Π(k,y) ∈ GnR such that for every fixed k
xy, if y ≤ k lth( )=0, lth( x0,...,xk )= k +1, Π(x, y)= if x = x0,...,xk . 00, otherwise 2For detailed information on this as well as various other codings see [33] and also [5] (where j is called ’Cauchy’s pairing function’). 3One easily shows that (x + y)2 + 3x + y is always even (This can be expressed as a purely universal sentence, i.e. ω 2 as an axiom in GnA ). Hence the case ’otherwise’ never occurs and therefore 2j(x,y) = (x + y) + 3x + y for all x,y.
10 Define
0 0 , if x =0 0 lth(x) := j1(x − 1)+1, otherwise,
0 0 , if lthx ≤ y y+1 Π(x, y)=0 j1 ◦ (j2) (x − 1), if 0 ≤ y < lthx − 1 (j )lthx(x), if lthx > 0 ∧ y = lthx − 1 2
We usually write (x)y instead of Π(x, y). ω In order to verify that Π(x, y) is definable in G2R it suffices to show that the variable iteration y ω ϕxy = (j2) (x) of j2 is definable in G2R . This however follows from the fact that ϕxy ≤ x for all x, y. Thus we can define ϕxy by R˜ using λy.x as bounding function. ω For n ≥ 3 we can code initial segments of variable length of a function f in GnA , i.e. there is a ω 4 functional Φ ∈ G3R such that Φ fx = f0,...,f(x − 1) : As an intermediate step we first show the definability of
˜ f0= f0 fx˜ ′ = ˜j(fx,˜ fx′), where ˜j(x, y) := j(y, x) x ω 2 3x M 2 in G3R : One easily verifies (using j(x, x) ≤ 4x ) that fx˜ ≤ 4 f x for all x. Hence the ′ x′ 3x M ′ 2 ω definition of f˜ can be carried out by R˜ using λx.4 f x ∈ G3R as bounding function. fx˜ means ˜j(. . . ˜j(˜j(f0,f1),f2) ...fx). Hence fx := (λy.f(x − y))x has the meaning ω j(f0,...j(f(x − 2), j(f(x − 1), fx)) . . .). We are now able to define Φ ∈ G3R :
0 0 , if x =0 Φ fx := (fx)x +1, otherwise, where
x, if y =0 fxy := f(y − 1), otherwise. ω We usually write fx for Φ fx. Furthermore one can define a function ∗ in G3R such that